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In order to answer a recent question on math.SE I had to look at special kinds of ultrafilters on the set of natural numbers. This question is of interest to me because it appeared in my own work on amenability several times already (e.g. in unpublished joint work with Vadim Kaimanovich).
A medial limit is a special kind of Banach limit with the additional property (among others) that its restriction to the unit ball of l^\infty is universally measurable (with respect to the weak* Borel structure, of course). See my answer on math.SE for more details.
It is known that it is a consequence of the continuum hypothesis (Martin's Axiom suffices) that medial limits exist. A natural question to ask is the necessity of assumptions beyond ZFC. I'd like to know for sure that going beyond ZFC is necessary for this. Martin's Axiom is far too much to require, but it seems that the precise question I'd like to ask is open, quoting Fremlin's measure theory (last paragraph on page 281 of Volume 5 I):
For most of the classes of filter [sic] here, there is a question concerning their existence. Subject to the continuum hypothesis, there are many Ramsey ultrafilters, and refining the argument we find that the same is true if $\mathfrak{p = c}$ (538Yb). There are many ways of forcing non-existence of Ramsey ultrafilters, of which one of the simplest is in 553H below. With more difficulty, we can eliminate $p$-point ultrafilters (Wimmers 82) or rapid filters (Miller 80) or nowhere dense filters and therefore measure-centering ultrafilters (538Hd, Shelah 98). It is not known for sure that we can eliminate medial limits or measure-converging filters (538Z).
Now the last sentence indicates that this was still considered an open problem by an eminent expert at most ten years ago. I'd like to ask a question (giving details on the definition I'm interested in of course) whether any progress has been made in that respect. I'm not very well acquainted with forcing so that it is a sheer impossibility for me to check myself or even recognize relevant results in the literature. On the other hand I see little point in asking about a problem that is known to be open, so I'd like to hear the community's assessment before posting a question on the main site that is likely to be closed as "too localized" or "off topic".
My hope is that the MO community would be able to provide me with either a definitive answer or valuable pointers to the literature.
Another and maybe even a better solution to this problem would be to contact Fremlin directly, of course.
Questions of the type "Is this problem still open" or "What recent progress has there been on conjecture X" are usually well received on MO. If the question is well written and gives sufficient background, then people who didn't know about the conjecture in the first place will already benefit from the question itself, even before an answer appears. And besides, there is just no other means of answering such questions. It would really take a leading expert in the field to say "there has been no progress whatsoever" (here is a personal example), and asking at MO is much gentler and less intrusive on the said expert than to email him.
Thank you so much Andreas, that's even better than what I was hoping to hear! I wonder how I managed to miss this paper in my rather extensive Google searches... Thank you also for your advice, Alex, I'll definitely bear that in mind in the future.
It seems you guys should ask and answer on the main site, just for the historical record. Who knows when I'll accidentally delete the entire meta site...
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